Higher algebraic $K$-theories

Abstract:

A homotopy fibration is established relating the Volodin or BN-pair definition of algebraic K-theory to the theory defined by Quillen. In [2] we outlined the construction of natural homomorphisms \[ K_ \ast ^Q \to K_ \ast ^{BN} \to K_ \ast ^V \to K_ \ast ^{KV}\] between higher algebraic K-theories $K_ \ast ^Q$ of [10] and [11], $K_ \ast ^{BN}$ of [17], $K_ \ast ^V$ of [16], and $K_ \ast ^{KV}$ of [7] and [8]. This was one of the steps in proving the various definitions of higher K-theory are equivalent. It turns out they all agree-including the theory $K_ \ast ^S$ of [14], [5], and [8]-provided one restricts to the category of regular rings when using $K_\ast ^{KV}$. See [1], [2], [5], [8] and [18]. The purpose of this paper is to prove the following theorem, announced in [2], which yields the construction of $K_ \ast ^Q \to K_ \ast ^{BN}$. Theorem. For any associative ring with identity A \[ G{L^{BN}}(A) \to B{\{ {U_F}\} ^ + } \to BGL{(A)^ + }\] is a homotopy fibration. For the reader’s convenience and because the presentation of the BN-pair K-theory $K_ \ast ^{BN}$ used here is slightly different from that of [17], we shall briefly recall the definition of $G{L^{BN}}$ and $B\{ {U_F}\}$ in the first section.